## How to merge two disjoint random samples?

The problem: Given two random samples, s1 and s2, of size k over two disjoint populations, p1 and p2, how to combine the two k-sized random samples into one k-sized random sample over p1 ∪ p2?

The solution: k times, draw an element s1 ∪ s2; with probability d1 = |p1| / |p1 ∪ p2|, draw the next element from p1; with probability d2 = 1 - d1 draw the next element from p2.

(the solution was on stackoverflow)

In python:

```import random import numpy   # sizes e1 = 1000 e2 = 1000000   # populations p1 = xrange(e1) p2 = xrange(e1, e2)   # sample size k = 500   # random samples s1 = random.sample(p1, k) s2 = random.sample(p2, k)   # merge samples merge = [] for i in range(k): if s1 and s2: merge.append(s1.pop() if random.random < len(p1) / float(len(p1)+len(p2)) else s2.pop()) elif s1: merge.append(s1.pop()) else: merge.append(s2.pop())   # Validate hist = numpy.histogram(merge, bins=[0,500000,1000000]) # The two bins should be roughly equal, i.e. the error should be small. print abs(hist - hist) / float(k)   # alternatively, use filter to count values below 500K print abs(len(filter(lambda x: x<500000, merge)) - 250) / 500.0```

## Non-strict orders: ≤

The symbol ≤ denotes a generalization of "less than or equal", and it defines either a partial or total ordering over a set P (in the table below a,b ∈ P):

Constraint (Non-strict) partial order (Non-strict) total order
Reflexivity: a ≤ a x x
Antisymmetry: if a ≤ b and b ≤ a then a = b x x
Transitivity: if a ≤ b and b ≤ c then a ≤ c x x
Totality: either a ≤ b or b ≤ a x

## Strict orders: <

The symbol < denotes a generalization of "less than", and it defines either a partial or total ordering over a set P (in the table below a,b ∈ P):

Constraint (Strict) partial order (Strict) total order
Irreflexivity: ¬(a < a) x x
Asymmetry: if a < b then ¬(b < a) x x
Transitivity: if a < b and b < c then a < c x x
Totality: either a < b or b < a x

Note the difference between asymmetry and antisymmetry.

Type of relation Constraint
Asymmetric relation if a < b then ¬(b < a)
Antisymmetric relation
(two equivalent definitions)
if a ≤ b and b ≤ a then a = b
if a ≠ b then ¬(b ≤ a)

## Metrics cheat sheet

Question: When can a distance function `d(x,y)` be called metric, pseudo-metric, quasi-metric or semi-metric?

Constraint Metric Pseudo Quasi Semi
Non-negativity: d(x,y) ≥ 0 x x x x
Identity of indiscernibles: d(x,y)=0 ⇒ x=y x x x
Symmetry: d(x,y) = d(y,x) x x x
Triangle inequality: d(x,z) ≤ d(x,y)+d(y,z) x x x

Table derived from Wikipedia article on metric spaces: http://en.wikipedia.org/wiki/Metric_(mathematics)